What is a partial derivative?

A partial derivative measures how a multivariable function changes when you vary one input while holding all others constant. For f(x,y), the partial with respect to x treats y as fixed and differentiates with respect to x only, giving the instantaneous rate of change in the x direction at a specific point.

How is a partial derivative calculated numerically?

This calculator uses the central difference formula: df/dx is approximately (f(x+h, y) minus f(x-h, y)) divided by 2h, where h is 1e-7. Central difference is more accurate than forward or backward difference because errors cancel symmetrically, giving around 12 significant figures for smooth functions.

What is the gradient vector?

The gradient vector of f(x,y) at a point is (df/dx, df/dy). It points in the direction of steepest ascent on the surface z = f(x,y). Its magnitude is the rate of change in that steepest direction. Gradients are used in optimisation and machine learning to find minima and maxima.

Partial Derivative Calculator

A partial derivative tells you how a function of two variables changes when you move along just one of those variables while holding the other fixed. If you have a surface defined by z = f(x, y), the partial derivative with respect to x gives you the slope of that surface in the x direction at a specific point, and the partial derivative with respect to y gives the slope in the y direction. Together they form the gradient vector, which points in the direction of steepest ascent on the surface. Partial derivatives appear throughout multivariable calculus, mathematical physics, engineering optimisation, and machine learning, where gradients are used to train models by descending a loss surface. This calculator takes a function of x and y written in standard notation, for example x^2 + 3*x*y, and a point (x, y) at which to evaluate it. It returns df/dx, df/dy, the gradient vector as an ordered pair, and the mixed second partial d2f/dxdy. Calculations use central difference numerical differentiation, which is accurate to around twelve significant figures for smooth functions. You can use standard operators: +, -, *, /, and ^ for powers, along with Math functions such as sin, cos, exp, log, sqrt, and abs. Enter the function and the point below to see all four results instantly.

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Function f(x, y)

x value

y value

7.00

df/dx at (x, y)

df/dy6.00

Gradient vector(7.00, 6.00)

d2f/dxdy3.00

f(x, y)10.00

Numerical results use central difference with h = 1e-7. Use * for multiplication and ^ for powers.

How it works

The calculator evaluates your function using JavaScript's Function constructor, which allows standard Math operations without any external library. The partial derivative with respect to x uses central difference: df/dx is approximately (f(x+h, y) minus f(x-h, y)) divided by 2h, with h = 1e-7. The same method applies in the y direction to find df/dy. The mixed partial d2f/dxdy is found by applying the central difference formula in x to the df/dy values evaluated at x+h and x-h. The gradient vector collects (df/dx, df/dy) into an ordered pair, and f(x, y) is the function value at the given point.

Worked example

For f(x, y) = x^2 + 3*x*y at the point x = 2, y = 1: differentiating with respect to x gives df/dx = 2x + 3y, which at (2, 1) equals 2(2) + 3(1) = 7.00. Differentiating with respect to y gives df/dy = 3x, which at (2, 1) equals 3(2) = 6.00. The gradient vector is (7.00, 6.00), the mixed partial d2f/dxdy = 3.00, and f(2, 1) = 4 + 6 = 10.00. These match the defaults pre-filled above.

Related calculators

Tangent Line Calculator: find the equation of the tangent to a single-variable function at a point.
Critical Points Calculator: locate where a function's derivative equals zero.
Average Rate of Change Calculator: compute the slope of a secant line over an interval.
Riemann Sum Calculator: approximate the area under a curve using rectangles.

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